Seasonality in the statistics of surface air temperature and the

Meteorol. Appl. 00, 1–10 (2003)
DOI:10.1017/S1350482703001105
Seasonality in the statistics of surface air
temperature and the pricing
of weather derivatives
Stephen Jewson1 & Rodrigo Caballero2
1
Risk Management Solutions, 10 Eastcheap, London, EC3M 1AJ, UK
Email: [email protected]
2
Department of the Geophysical Sciences, University of Chicago, 5734 S. Ellis Ave.,
Chicago IL 60637, USA
Email: [email protected]
The pricing of weather derivatives motivates the need to build accurate statistical models of daily
temperature variability.Current published models are shown to be inaccurate for locations that show
strong seasonality in the probability distribution and autocorrelation structure of temperature
anomalies. With respect to the first of these problems, we present a new transform that allows seasonally
varying non-normal temperature anomaly distributions to be cast into normal distributions. With
respect to the second, we present a new parametric time-series model that captures both the seasonality
and the slow decay of the autocorrelation structure of observed temperature anomalies. This model is
valid when the seasonality is slowly varying. We also present a simple non-parametric method that is
accurate in all cases, including extreme non-normality and rapidly varying seasonality. Application of
these new methods in some realistic weather derivative valuation examples shows that they can have a
very large impact on the final price when compared to existing methods.
1. Introduction
Weather derivatives are contracts that allow entities to
insure themselves against the financial losses that can
occur due to unfavourable weather conditions. For
instance, there are many retailing companies which
have sales that are adversely affected by either warmer
or colder than normal temperatures. These companies
could all potentially benefit from hedging their exposure
with temperature-based weather derivatives.
The payoff from a weather derivative is determined by
the outcome of a weather index such as mean summer
or winter temperature. Pricing of weather derivatives is
mainly based on estimates of the distribution of possible
values of the index. Before any forecasts are available
historical weather data must be obtained to make this
estimate. Commonly a simple method is used in which
the values of the index for past years are determined.
Using 30 years of historical data would give 30 historical
index values (for example, 30 mean winter temperatures)
and these would be taken as 30 independent samples
from the distribution to be estimated. In most cases, a
trend would be removed prior to estimating the index
distribution, either from the daily weather data or from
the 30 annual historical index values. This method for
deriving the index distribution is known as (detrended)
burn analysis.
The most obvious disadvantage of burn analysis is that
it does not sample the possible extreme outcomes of
the index very well. This can be overcome by fitting an
appropriate distribution to the historical index values,
which smooths the distribution and extrapolates it to
higher and lower levels of probability. This is known as
index modelling.
However, neither of these so-called index-based
approaches for determining the distribution of the index
achieve the highest possible accuracy because much of
the historical data is discarded in the calculation of the
historical index values. For example, when considering
a one-week contract, 51/52 of the data is not used by
these methods. If some of the data from the other 51
weeks of the year contain statistical information about
the behaviour of weather during the one week of the
contract, as is likely, then it could be more accurate to
use some of these data too.
These considerations lead one naturally to consider the
possibility of using statistical modelling of temperatures
on a daily basis, and considerable work has gone
in to trying to build such models. In the most
common approach, a deterministic seasonal cycle is
removed from the mean and/or standard deviation
of temperature, and the residuals (which we will
call temperature anomalies) are then modelled using
1
Stephen Jewson & Rodrigo Caballero
continuous or discrete stochastic processes. The difficulty lies in finding stochastic processes that accurately
capture the observed behaviour of the temperature
anomalies. Dischel (1998), Cao & Wei (2000), Torro
et al. (2001) and Alaton et al. (2002) have suggested using
AR(1) (first-order autoregressive) models or continuous
equivalents. Others (Dornier & Querel 2000; Diebold &
Campbell 2001) have suggested more general models
that all lie within the larger class of autoregressive moving-average (ARMA) models (Box &
Jenkins 1970). Caballero et al. (2002) (henceforth CJB)
have shown that all these models fail to capture the
slow decay of the temperature autocorrelation function,
and hence lead to significant underpricing of weather
options. CJB and Brody et al. (2002) (henceforth
BSZ) have suggested Gaussian discrete and continuous
stochastic processes, respectively, that overcome this
problem by using models with long memory (i.e.
power law decay of the autocorrelation function).
CJB model the variability of temperature anomalies
using a stationary autoregressive fractionally integrated
moving average (ARFIMA) process (Granger & Joyeux
1980), while BSZ use a fractional Ornstein-Uhlenbeck
(fOU) process. ARFIMA and fOU models are mutual
analogies in the discrete and continuous domains. These
models work well for locations where the assumptions
of normality and stationarity are accurate. However,
as we shall see below, temperature anomalies at many
locations show marked departures from normality and
stationarity. In these cases, the CJB and BSZ models
should not be used for pricing weather derivatives as
they are likely to give misleading results.
This paper tackles the question of how to model
temperatures at these locations. In Section 2 we describe
the data sets to be used. In Section 3 we examine
the evidence for non-stationarity in the autocorrelation
structure of surface air temperatures. In Section 4 we
model it using a new class of parametric statistical
models that we have developed specifically for this
purpose. In Section 4.2 we also describe a simple
non-parametric model that provides an alternative
method for cases where parametric models entirely fail.
In Section 5 we investigate seasonally varying nonnormality of temperature variability and we present
a new method that can be used to model such nonnormality. In Section 6, we compare the performance
of the various approaches in pricing a specific weather
derivative. In Section 7 we draw some conclusions.
2. Data
The studies of temperature variability described in this
paper are based on two data sets. The first data set
originates from the US National Weather Service and
consists of daily minimum and maximum temperatures
measured between midnight and midnight (local time)
for 40 years (1961–2000) at 200 US weather stations (in
this study we only present results for eight of them,
2
Table 1. The eight US weather stations used in the modelling
described in the text, with the optimum lengths of the four
moving averages as selected automatically as part of the fitting
procedure for the AROMA model.
Station
Chicago Midway
Miami
Los Angeles
Boston
New York Central Park
Charleston
Detroit
Atlanta
m1
m2
m3
m4
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
3
4
9
5
4
4
3
7
17
28
33
32
18
22
24
27
listed in Table 1). These data are not directly suitable for
analysis because of (a) gaps in the data due to failures
in measurement equipment, recording methods or data
transmission and storage, and (b) jumps in the data
due to changes in measurement equipment or station
location. To avoid these problems we use data that
have been processed by Earth Satellite Corporation to
fill in such gaps and remove such jumps (Boissonnade
et al. 2002). These processed data were provided to the
authors by Risk Management Solutions Ltd. Even after
such processing the data are still not representative of
the current climate because of trends due to urbanisation
and global warming. Removing such trends is extremely
difficult, as the trends vary by location, by year and
by season. Attempting to model the variations by
year and season can only by done rather subjectively,
and so we have restricted ourselves to the simple and
transparent approach of removing a non-seasonal linear
trend, estimated over the entire data period, for daily
mean temperature at each station. The linear trends were
fitted by the method of least squares, and removed in
such a way that the detrended data values are consistent
with the last day of the data (in other words, where
there is a warming trend we increase the earlier values up
to present-day levels). Most weather contracts depend
on daily mid-point temperature, calculated as the midpoint between the daily maximum and minimum, and
so all our modelling will focus on these values.
The second data set is the US National Centre for
Environmental Prediction (NCEP)’s 40 year reanalysis
(Kalnay et al. 1996). This data set is obtained by
assimilating observations into a dynamical model and
produces a gridded representation of the climate over
the whole globe. We only use surface temperature values
from these data. The reason we use these data in addition
to the station data described above is simply that the
reanalysis data are conveniently presented on a spatial
grid, which makes it much easier to produce maps of
spatial fields.
Both temperature data sets are converted to anomalies
by removing deterministic seasonal cycles in the mean
Seasonality and the pricing of weather derivatives
Figure 1. Observed ACFs for Chicago (top row) and Miami
(bottom row) in winter (defined as December–January, DJF)
and summer (June–August, JJA). In each panel, the solid black
line is the annual ACF (i.e. the ACF computed using the entire
data set) and the dotted line is the ACF for the season specified
(computed as the average of the seasonal ACFs in the 40
individual years). The dashed lines show the 95% confidence
intervals around the observed estimate, calculated using the
method of Moran (1947).
and the standard deviation: this was achieved by
regression onto three sines and cosines in each case.
3. Seasonality in the autocorrelation structure
of surface air temperatures
In Figure 1 we compare the seasonal autocorrelation
functions (ACFs) at two locations, Chicago and
Miami. Chicago shows essentially no seasonality,
with no statistically significant difference between
summer, winter and annual ACFs. The situation is
strikingly different in Miami. Persistence of temperature
anomalies is clearly much higher in summer than in
winter. It is clear from this that a stochastic model (such
as those used in CJB or BSZ) that assumes stationarity
of the ACF will severely underestimate the memory in
summer, and will overestimate it in winter. If such a
model is used to derive the distribution of a weather
index such as cumulative temperature, it will severely
underestimate the standard deviation of the index in
summer (see CJB, Equation 11)
We turn to the NCEP data set to examine how
widespread seasonality of the ACF is. We define an
index s as
s=
40
ρk
(1)
k=1
where ρk is the ACF for a particular season at lag k.
The higher the value of s , the greater the persistence
of the temperature anomalies. Figure 2 shows maps
of s for summer and winter over North America and
Europe. In both seasons, persistence is greater over
the oceans. In winter, persistence is relatively uniform
over North America, but shows a banded pattern over
Europe, with high-persistence bands over Scandinavia
and the Mediterranean and lower persistence over the
southern European mainland. The situation changes
markedly in summer. The biggest change is over the
oceans, where persistence increases almost everywhere.
There are also changes over the continents. In North
America, persistence increases over the Gulf states
and southwestern USA, but decreases over coastal
California. In Europe, the banded pattern described
above is still in place but the relative amplitudes change,
with persistence decreasing over Scandinavia and central
Europe and increasing over the Mediterranean. In
summary, we find strong seasonality of the ACF over
large (and economically significant) parts of the USA
and Europe. This motivates the search for a time series
model capable of capturing such seasonality.
4. Modelling of seasonality in the
autocorrelation structure
We now proceed to the main topic of this article, which
is the modelling of seasonality in the autocorrelation
structure of temperature. A first approach might be to
try to extend the ARFIMA model of CJB to include
seasonality. This could be attempted by allowing the
parameters of the model to vary with time of year, or
perhaps by fitting the model to data from only one
part of the year. Both such approaches are theoretically
possible. However, they are also rather complex. The
ARFIMA model represents the slow decay of the ACF
using a long memory parameter d. A model that allows
the long memory parameter d to vary with the time of
year is hard to fit, and fitting d on data for only one
season is also difficult.
CJB suggested that the long memory of surface air
temperatures may simply result from the aggregation
of several processes with different time-scales, such as
internal atmospheric variability on short time-scales,
land-surface processes on medium time-scales, and the
interaction of the atmosphere and ocean on long timescales. Indeed, a simple statistical model consisting of
a sum of 3 AR(1) processes was shown to give results
that were indistinguishable from long memory over the
length of data and number of lags used. This 3 × AR(1)
model is not, however, practical for simulating artificial
temperatures because the parameters cannot be easily
estimated. It does, nevertheless, motivate the search
for other simple models that might perform as well as
ARFIMA and yet avoid the difficulties introduced by
the long memory parameter d.
It was shown in CJB, and subsequently in more detail
in Brix et al. (2002) that the well-known ARMA
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Stephen Jewson & Rodrigo Caballero
Figure 2. Persistence of surface air temperature anomalies as quantified by the index s (defined as an average of the ACF at each
point over the first 40 lags; see text) in (a) winter (DJF) and (b) summer (JJA). Contour interval is 0.05. Light shading shows
values > 0.1, dark shading values > 0.2. (c) Difference (DJF−JJA). Contour interval is 0.05. Dashed lines show negative values.
Shading shows absolute values > 0.05. Data from NCEP Reanalysis, 1950–1999.
processes (Box & Jenkins 1970) do not work well for
surface temperature anomalies because they do not
capture the observed slow decay of the ACF. For
an AR process to capture this shape of ACF out
to 40 lags, 40 parameters are needed. Such a model
is extremely non-parsimonious, and the parameters
cannot be estimated with any accuracy. Indeed, it would
be impossible to distinguish most of the parameters
from zero. This is unfortunate, because it would be
relatively straightforward to generalise the AR process
to have seasonally varying parameters, and hence solve
the problem of modelling seasonality.
We now present a new statistical model that maintains
the simplicity of the AR processes, but is as accurate and
parsimonious as the more complex ARFIMA model.
Initially, we present a non-seasonal version of the model,
but extend it to include seasonality in Section 4.1. The
model is patterned after AR( p):
xn = α1 xn−1 + α2 xn−2 + · · · + α p xn− p + ξn ,
(2)
where xn is the value of the process at step n, ξn is
a Gaussian white-noise process, and αi are constants,
but rather than using individual temperatures for days
n − 1, n − 2, . . . as predictors, we use moving averages
4
Figure 3. Averaging intervals used in the AROMA model.
of past temperatures:
xn = α1 ym1 + α2 ym2 + · · · + αr ymr + ξn ,
(3)
where
ym =
m
1 xn−i
m i=1
(4)
Note that all the averages start from day n −
1. This is sketched graphically in Figure 3. Since
this model consists of autoregressive terms on
moving averages, we give it the name AROMA. An
AROMA(m1 , m2 , . . . , mr ) process can be rewritten as
an AR(mr ) process, but it can accurately capture the
observed temperature autocorrelation structure with
much fewer than mr independent parameters (see
below).
Seasonality and the pricing of weather derivatives
readily to include seasonality. From the standpoint
of meteorological interpretation, AROMA is more
attractive than the ARFIMA model, appealing as it
does to the idea of different timescales in weather
variability, corresponding to each of the moving average
terms. If temperature today is related to an average of
temperature over the last two days, then this is probably
just a reflection of the impact of small scale weather
systems. If it is related to an average of temperatures
over the last 20 days then this may be because of the
memory in soil moisture, for instance.
4.1. Extending AROMA to include seasonality
Figure 4. The observed (solid line) and modelled ACFs
for Chicago. The modelled ACFs were produced using an
ARFIMA model (dotted lines) and the AROMA model
(circles).
How are we to choose the number and length of
moving averages to use in the model, and calculate the
coefficients αi ? As for the number of moving averages,
this should be chosen to be as small as possible, so that
the parameters can be well estimated. Experiments on
temperature anomalies for our 8 stations suggest that
four moving averages (i.e. r = 4) are typically enough
to capture the observed ACF well out to 40 lags: this
is a great improvement on the number of parameters
required by the AR model for the same accuracy. Given
values for m1 , m2 , m3 and m4 , it is straightforward to
calculate the weights α1 , . . . , α4 using linear regression.
All that remains is to decide which moving averages to
use. Experiments were performed on our eight stations
in which all combinations of values of m1 , m2 , m3 and m4
up to 35 were tested. The results were ranked according
to the root mean square error between the model ACF
and the observed ACF (an alternative method would
be to rank the results using the likelihood of each
model). Results are shown in Table 1. Interestingly,
all locations were modelled optimally using m1 = 1 and
m2 = 2. Values of m3 and m4 , were, however, different
for different stations. This suggests that a simple way
to choose the lengths of the moving averages to be used
is to fix m1 = 1 and m2 = 2, and optimise the ACF by
varying the other two lags. This is a simple optimisation
problem and can be solved in a matter of seconds on a
personal computer.
Figure 4 shows the observed and modelled annual ACF
for Chicago using the AROMA and ARFIMA models.
We see that both the AROMA and the ARFIMA
models give a very good fit to the observed ACF.
The advantage of the AROMA model is that it can
be fitted much more simply (as we have seen above)
and (as we shall see below) can be extended very
We now extend the AROMA model to include
seasonality. First, we fit the model to annual data, as
described above. This fixes the lags m1 , m2 , m3 , m4 that
are to be used. We do not use different lags at different
times of year. For each day of the year, we then fit
a different model with different regression parameters
αi . This is done be selecting a window of data either
side of that day, and performing the regression using
data within that window. The regression parameter for
a moving average of length m can only be fitted if
the length of the fitting window is significantly larger
than m, in order that the window contain a sufficient
number of regression pairs that a sensible regression can
be performed. In our case, we limit the lengths of the
moving averages to 35 days, and set the window length
to 91 days (an ad hoc choice designed to give enough
data to allow accurate fitting of the model for each day
while still allowing for seasonal variability). Thus even
for the longest moving average of 35 days we have 66
pairs of values from each year of data with which to
calculate the appropriate αi . The data used for adjacent
days of the year overlap almost entirely, bar one day
at each end of the 91-day window, and so we would
expect the regression coefficients to vary only slowly
during the year.
Figure 5 shows observed and modelled ACFs for
different seasons for Miami. In each panel of this figure
we show the annual ACF for reference, which is the
same in all four figures. The solid line in each case
shows the observed ACF for that season. Because these
seasonal ACFs are calculated using fewer data than for
the annual ACF, they show more sampling variability
and are hence less smooth. The circles show the seasonal
ACFs from the seasonal AROMA (SAROMA) model
fitted to the Miami daily temperatures as described
above. We see that the model captures the seasonal
variation in the ACF well. In particular it shows a slow
decay of memory in summer, and more rapid decay in
the other seasons.
Figure 6 shows the seasonal variation of the four
regression parameters in the SAROMA model. We see
that they vary slowly with the time of year. The slow
decay of the ACF in summer corresponds to a summer
5
Stephen Jewson & Rodrigo Caballero
4.2. A non-parametric method: extended
burn analysis
Figure 5. The four panels show observed and modelled ACFs
for Miami for the four seasons. The observed data are the
same as in Figure 1. In each panel the dotted line is the
annual ACF which is included for reference. The solid line is
the observed ACF for that season, and the circles are the
modelled (SAROMA) ACF for that season. Confidence limits
are omitted for clarity.
Figure 6. Seasonal variation of the four regression parameters
for the SAROMA model for Miami. The solid lines show the
estimated parameter values, while the dotted lines show the
95% error bounds. We see that each of the parameters shows a
strong seasonal cycle, corresponding to the strong seasonal cycle
seen in the observed and modelled ACFs.
peak in the fourth parameter which applies to a moving
average of length 28 days. The SAROMA model could
be extended by smoothing these regression parameters
in time (either parametrically or non-parametrically) to
preserve only the long time-scale variations and remove
the short time-scale fluctuations which are presumably
only due to sampling error.
6
We have discussed non-seasonality in the ACF, and
shown that it is strong in some cases. We have also
introduced a simple parametric model that captures this
seasonality well. There are, however, potential limits
to how well the model can work. In particular, if the
timescales of change of the observed ACF are rapid, then
one would want to use a short window to fit the model.
A short window, however, prevents accurate estimation
of the parameters because the regression has insufficient
data. The model is thus limited to representing ACFs
that change over timescales somewhat longer than the
longest lag in the model. Another limitation is that the
model uses the same lags at different times of year, while
it could be that the timescales of memory in the physical
system actually change with time of year. Finally,
the model assumes that the (distribution adjusted)
temperature anomalies are governed by linear dynamics,
which may not be entirely correct. It is possible that one
could make a different parametric model that would
solve some of these problems in a satisfactory way.
However, given the richness and complexity of climate
variability, we believe it will always be the case that
there will be some location somewhere for which the
observed temperature variability will not fit a given
parametric form. Because of this, it would seem useful
to also explore non-parametric methods that make the
fewest possible assumptions about the data, and hence
are likely to be generally applicable.
We present here one such simple method that is
essentially an extension of the burn analysis described in
the Introduction. It relies on the simple idea that there
may be some information from outside the contract
period which is relevant to the contract period itself.
Consider for instance a one-week contract: we may
expect data from the weeks preceding and following the
contract period to have statistics similar to that of the
contract period itself. There is an implicit assumption
here that the ACF and distribution of variability vary
only fairly slowly (i.e. do not change much from week to
week), and that the inaccuracy introduced by the small
week-to-week changes in the ACF and distribution
can be outweighed by the benefits of having more
data to work with. Note, however, that this model can
work with ACFs that change more rapidly than the
SAROMA model can accommodate.
The method works as follows. We define a period that
extends either side of the contract period and captures
the data to be used. In the above example, we might
employ a window of two weeks on either side of
the contract period. This gives us five times as much
historical data to work with than using only the contract
period (as in standard burn analysis), which would be
expected to increase the accuracy of our estimates by
more than a factor of 2. We now slide a window of the
same length as the contract period along the relevant
Seasonality and the pricing of weather derivatives
data. For each window position, we add back the
seasonal cycle in variance and mean appropriate for the
contract period, and calculate a historical index value.
The end result is many more historical index values than
are obtained in the index-based methods. In our example
we have a seven-day contract and a 35-day relevant data
period which means that the seven-day window can
take 29 different positions. Each year of data thus gives
us 29 historical index values. This is 29 times as many
as if we only used the contract period itself (although
these 29 values are not independent). The advantage of
sliding the window rather than jumping it (even though
the underlying data used are the same) is that (a) it
creates a smoother estimate of the final distribution,
and (b) it uses more possible combinations of days of
daily weather, which can result in more extreme values
for the index.
The only remaining issue is to decide the length of
the relevant data period. Too long a period will start
including data that are not relevant because the statistics
of weather variability have changed. Too short a period
will not reap the potential benefits of using more data.
The optimum window length is clearly dependent on
the station: with Chicago we might be tempted to use
data for the whole year, while with Miami that would
clearly be wrong since the summer data are markedly
different from those of the other seasons. One way
to choose the window length would be to analyse
seasonal variations in higher moments of temperature
variability and seasonal variations in autocorrelations
at, or averaged over, certain lags.
5. Seasonal non-normality of surface
air temperatures
By construction, our temperature anomalies are
stationary in the mean and the variance. It is still
possible, however, that they are non-stationary in higher
moments of variability. Figure 7 shows the cumulative
distribution functions (CDFs) of observed temperature
anomalies for the four seasons for Miami. We have
plotted these distributions against a fitted normal
distribution in the form of a QQ plot. If the observed
data are normally distributed, they will lie along the
diagonal. If they are skewed they will tend to lie at an
angle to, and cross, the diagonal. If there are heavy tails at
the warm end of the distribution, the data will lie below
the diagonal and if there are heavy tails at the cold end
of the distribution, the data will lie above the diagonal.
We see that all seasons show deviations from a normal
distribution with light warm tails and heavy cold tails.
The largest deviations are the light warm tails in summer,
showing that extreme warm events are much less likely
than would be supposed from a normal distribution.
The levels of non-normality seen in many locations
are sufficiently large that making the mathematically
convenient assumption of normality will degrade the
Figure 7. The four panels show QQ plots for temperature
anomalies in Miami for the four seasons. The horizontal axis
shows the observed quantiles, while the vertical axis shows the
modelled quantiles. We see that in all seasons the cold tail of
the distribution is heavy tailed (cold events are more likely
than predicted by the normal distribution) while the warm tail
of the distribution is light tailed (warm events are less likely
than predicted by the normal distribution). The most significant
departure from normal is the warm tail in winter.
accuracy of the temperature simulations significantly,
and lead to mis-pricing of weather derivatives. The
errors will be particularly large for contracts which
depend heavily on extreme values of temperature, but
will also be important for standard contracts. It is thus
sensible to attempt to model this non-normality directly.
There are a number of possible methods that can be
used to model non-normality in temperature variability.
An initial observation we make is that although
temperatures are clearly not normally distributed at
many locations, the distribution is, at the same time,
still reasonably close to normal. This suggests that if
we approach the problem using transformations that
convert the data to a normal distribution, then the
dynamics will not be affected too strongly, and the same
time series models as are used in the normal case may
still work.
The first choice is whether to model the non-normality
before or after attempting to model the autocorrelation.
The first of these methods would involve applying
a transformation to the temperature anomalies that
renders them more or less normal. Time series modelling
methods that assume normality, such as ARFIMA or
SAROMA, can then be applied to the transformed
anomalies. Simulated values are passed back through the
transformation to get back to the original distribution.
7
Stephen Jewson & Rodrigo Caballero
A second method would be to apply a time series model
to the temperature anomalies directly, and model nonnormality in the residuals. We choose to apply the first of
these methods because it allows us to re-use algorithms
such as the ARFIMA or SAROMA models simply by
applying a transformation to the inputs and outputs of
those models.
The second choice is whether to use parametric or
non-parametric models for the transformation of the
anomalies. Box–Cox transformations (Box & Cox 1964)
are a commonly used parametric distribution transform,
and can be extended to vary seasonally. They are not,
however, completely general. Non-parametric methods,
on the other hand, can cope with any temperature
anomaly distribution. For this reason we choose a nonparametric method, and the method we present, is, as
far as we are aware, new.
The method works as follows. We derive a separate
estimate of the cumulative distribution of the temperature anomalies for each day of the year (this step is
discussed in detail below). These cumulative distributions are used to convert the temperature anomaly
for each day (over all years) into a probability, and
that probability is then converted, using the inverse of
the standard normal CDF, into a value sampled from a
normal distribution.
It remains to specify how to estimate the distributions
used for each day. They could be estimated using only
data from that day of the year: however, this would give a
poor estimate because using 50 years of data would only
give 50 points on the distribution. Instead we assume
that the anomaly distribution only varies slowly with
time and that we can estimate this distribution more
accurately by using additional data from surrounding
days. We do this by taking temperature values from
a window surrounding the actual day, with a window
length of 91 days (arbitrarily chosen to be long enough
to give a smooth distribution, but short enough not to
smooth out seasonal variations). Thus for each day of the
year, we form the estimate of the anomaly distribution
for that day using 91 days per year × 50 years = 4550
days of data. This gives a smooth estimate. Since the distributions for adjacent days are based on almost the same
data because of substantial overlapping of the windows,
the transform only changes gradually with the time of
year, which seems realistic.
The results of the application of this transform to
Miami temperatures are shown in Figure 8. We see
that most of the non-normality has been removed.
The transformed temperatures can now be modelled
using a Gaussian process, and simulations of the
transformed temperatures can be converted back to
the appropriate distribution using the inverse of the
distribution transform. One caveat for this method is
that as long as we use an empirical distribution as
described above, it would not be possible for final
8
Figure 8. The four panels show QQ plots for temperature
anomalies in Miami for the four seasons, after having been
transformed using the non-parametric seasonally varying
transform described in the text. We see that most of the nonnormality has been removed.
simulated temperature anomalies to exceed historical
values. This is unlikely to be a problem in most
cases. However, if we have a particular interest in
extreme events then it would be advisable to extend the
distribution used in the transformation using extreme
value theory (Reiss & Thomas 1997).
6. Examples
We now present some examples to illustrate the
effects of improved modelling of the seasonality in the
distribution (Section 5 above) and the ACF (Section 4
above) on the calculation of weather derivative prices.
Our examples are all based on Miami, since that location
has strong levels of seasonality in both the distribution
and the ACF and so is likely to show the greatest benefits
of using the more advanced modelling methods.
The type of weather derivative we consider for our
examples are ‘unlimited call options’. These require
one party (the seller of the option) to pay another
(the buyer) a certain amount of money if the final
weather index is above a specified level known as the
strike. The amount paid (the payoff) is proportional to
the difference between the index and the strike, with
a constant of proportionality known as the tick. This
effectively insures the buyer against high values of the
index. The question is: given the strike and tick, what
should the premium (i.e. the price paid by the buyer)
be? The simplest answer is that the premium should
equal the expected payoff of the contract. Then, in the
Seasonality and the pricing of weather derivatives
Table 2. The contract structures for the examples.
Strike
Tick
Summer contract
Winter contract
1750 CDDs
$5000/CDD
90 HDDs
$5000/HDD
long run, neither the buyer nor the seller will make
profit. In practice, the seller may add a risk loading on
top of the premium to compensate for the risk taken in
underwriting the derivative. For the present discussion
we will ignore this issue and focus on the estimation of
the expected payoff. Given the payoff structure and the
index distribution, we can easily estimate the expected
payoff using numerical integration. We will do this for
both ARFIMA and SAROMA models, both with and
without the distribution transform (giving four cases in
all). We will consider two examples: a winter contract
(December to February) and a summer contract (June
to August).
The details of these contracts are shown in Table 2.
The winter contract is based on heating degree days
(HDDs), which are defined as the sum of the excursions
below 18 ◦ C (65◦ F) during the contract period, while
the summer contract is based on cooling degree
days (CDDs) which are defined as the sum of the
excursions above 18 ◦ C during the contract period (these
definitions are standard in the energy industry). HDDs
are a measure of how cold the season is, and relate to
use of heating; CDDs are a measure of how warm the
season is and relate to the use of cooling.
Table 3 shows the results for the summer contract.
The index means for the different models are virtually
identical in all cases. This is because summer in Miami
is very warm and the temperature rarely drops below
18 ◦ C. As a result, the mean number of CDDs is fixed
by the seasonal cycle (see Eq. 10 in CJB) which is
modelled in the same way for all the models presented.
The small differences are due to sampling error, caused
by the use of a finite number of years of simulation.
The differences between the models appear in the index
standard deviations. The ARFIMA model gives lower
standard deviations that the SAROMA model, for both
normal and transformed distributions. This is explained
by the higher autocorrelation in summer (see Figure 5),
which is captured by SAROMA but not by ARFIMA.
Higher autocorrelations directly lead to a higher index
standard deviation (see Eq. 11 in CJB). As a result,
SAROMA prices are over 25% higher than ARFIMA
prices. A seller using ARFIMA pricing would, on
average, lose over $20,000 on this contract.
In the summer example above, it makes little difference
whether one uses a normal or transformed distribution
in the models: clearly, the distribution is always close
to normal in this case. Things are quite different in
the winter (Table 4). The mean index is now much
higher for the models with the distribution transform
than not. This can be explained as follows. Miami in
winter is quite warm, and temperatures below 18 ◦ C are
uncommon. It is only the cold tail of the temperature
distribution that creates HDDs, and hence modelling
of this cold tail is crucial. Without transforming the
temperature distribution, the cold tail is modelled as
being too thin, and fewer HDDs occur in the model
than in reality. This causes the normally distributed
models to underestimate the mean number of HDDs.
The transformed-distribution models are more accurate
since they take into account the fatter cold tail in winter
(Figure 7). This has a dramatic impact on the expected
payoff: prices given by the transformed-distribution
models are almost three times higher than those of
the normally-distributed models. In this case, it hardly
matters whether one used ARFIMA or SAROMA, since
most of the price difference is due to the change in the
index mean rather than its standard deviation.
Table 3. Results for the summer contract example. Expected payoff values have been rounded to three significant figures.
Model
Distribution
Index mean
(CDDs)
Index std. dev.
(CDDs)
Expected
payoff ($)
ARFIMA
ARFIMA
SAROMA
SAROMA
Normal
Transformed
Normal
Transformed
1727.1
1727.5
1727.0
1727.4
62.1
62.5
73.5
73.6
75,900
76,400
96,200
96,900
Table 4. Results for the winter contract example. Expected payoff values have been rounded to three significant figures.
Model
Distribution
Index mean
(HDDs)
Index std. dev.
(HDDs)
Expected
payoff ($)
ARFIMA
ARFIMA
SAROMA
SAROMA
Normal
Transformed
Normal
Transformed
56.1
85.2
57.1
87.1
45.1
61.0
45.8
60.7
15,000
43,200
16,100
44,300
9
Stephen Jewson & Rodrigo Caballero
Taken together, these examples emphasise the importance of modelling both the distribution and the
ACF of temperature correctly in the pricing of weather
options.
7. Summary
The advent of weather derivatives has created
significant interest in the understanding and statistical
modelling of surface air temperature variability. Weather
derivative pricing methods based on modelling of daily
temperatures have certain potential advantages over
simpler methods. Such modelling is not, however, easy
because of the richness and complexity of climate
variability, and in particular, because of long memory
and seasonality. The CJB and BSZ models were the
first to capture the observed slow decay of memory of
surface air temperature variability. They are, however,
limited by assumptions of normality and stationarity
and, as we have shown, many locations do not conform
to these restrictions.
We have presented a relatively simple framework
in which the non-normality and seasonality of
temperature variability can be accommodated, as long as
it is reasonably slow in varying: changes in probability
distribution and ACF from season to season can be
captured, but much more rapid changes cannot. The
model for seasonal variation in the ACF that we present
can also be interpreted more simply than the long
memory models. The latter capture the slow decay
of the ACF using a slightly mysterious long memory
parameter d. Our model, however, presents temperature
today as the sum of components of temperature variability on different timescales, some short (presumably
due to small scale weather variability), and some longer
(presumably due to either atmospheric long waves, or
soil or ocean processes). Finally, we present a nonparametric model that can be applied to the pricing of
weather derivatives in those cases where the parametric
methods presented still do not appear to give a good fit
to the observed variability.
The models we have presented should lead to the
more accurate pricing of weather derivatives, especially
for contracts based on locations that show strong
seasonality. Furthermore, since they deal with the
difficult problem of modelling non-normal time series
with slowly decaying autocorrelations and seasonally
varying dynamics in novel ways, they may find applica-
10
tions in other areas of geophysical and environmental
modelling.
Acknowledgments
The authors would like to thank Jeremy Penzer, Anders
Brix and Christine Ziehmann for useful discussions
on the subject of the statistical modelling of daily
temperatures. Rodrigo Caballero was supported by
Danmarks Grundforskningsfond.
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